286 research outputs found
Elliptic pfaffians and solvable lattice models
We introduce and study twelve multivariable theta functions defined by
pfaffians with elliptic function entries. We show that, when the crossing
parameter is a cubic root of unity, the domain wall partition function for the
eight-vertex-solid-on-solid model can be written as a sum of two of these
pfaffians. As a limit case, we express the domain wall partition function for
the three-colour model as a sum of two Hankel determinants. We also show that
certain solutions of the TQ-equation for the supersymmetric eight-vertex model
can be expressed in terms of elliptic pfaffians.Comment: 34 page
Pfaffians and Representations of the Symmetric Group
Pfaffians of matrices with entries z[i,j]/(x\_i+x\_j), or determinants of
matrices with entries z[i,j]/(x\_i-x\_j), where the antisymmetrical
indeterminates z[i,j] satisfy the Pl\"ucker relations, can be identified with a
trace in an irreducible representation of a product of two symmetric groups.
Using Young's orthogonal bases, one can write explicit expressions of such
Pfaffians and determinants, and recover in particular the evaluation of
Pfaffians which appeared in the recent literature.Comment: 28
A Schur function identity related to the (-1)-enumeration of self-complementary plane partitions
We give another proof for the (-1)-enumeration of self-complementary plane
partitions with at least one odd side-length by specializing a certain Schur
function identity. The proof is analogous to Stanley's proof for the ordinary
enumeration. In addition, we obtain enumerations of 180-degree symmetric
rhombus tilings of hexagons with a barrier of arbitrary length along the
central line.Comment: AMSLatex, 14 pages, Parity conditions in Theorem 3 corrected and an
additional case adde
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